Questions: Graph the given functions, (f) and (g), in the same rectangular coordinate system. Describe how the graph of (g) is related to the graph of (f). (f(x)=x) (g(x)=x-10) Use the graphing tool to graph the functions. How is the graph of (g) related to the graph of (f)? A. The graph of (g) is the graph of (f) shifted 10 units vertically up. B. The graph of (g) is the graph of (f) shifted 10 units vertically down. C. The graph of (g) is the graph of (f) shifted 10 units horizontally left. D. The graph of (g) is the graph of (f) shifted 10 units horizontally right.

Graph the given functions, (f) and (g), in the same rectangular coordinate system. Describe how the graph of (g) is related to the graph of (f).

(f(x)=x)
(g(x)=x-10)

Use the graphing tool to graph the functions.

How is the graph of (g) related to the graph of (f)?
A. The graph of (g) is the graph of (f) shifted 10 units vertically up.
B. The graph of (g) is the graph of (f) shifted 10 units vertically down.
C. The graph of (g) is the graph of (f) shifted 10 units horizontally left.
D. The graph of (g) is the graph of (f) shifted 10 units horizontally right.

Transcript text: ph the given functions, $f$ and $g$, in the same rectangular rdinate system. Describe how the graph of $g$ is related to graph of $f$. \[ \begin{array}{l} f(x)=|x| \\ g(x)=|x|-10 \end{array} \] the graphing tool to graph the functions. $w$ is the graph of $g$ related to the graph of $f$ ? A. The graph of $g$ is the graph of $f$ shifted 10 units vertically up. B. The graph of $g$ is the graph of $f$ shifted 10 units vertically down. C. The graph of $g$ is the graph of $f$ shifted 10 units horizontally left. D. The graph of g is the graph of f shifted 10 units horizontally right.

Solution

Solution Steps

Step 1: Graphing the function $f(x) = p(x)$

To graph the function $f(x) = p(x)$, plot the points of $p(x)$ on a rectangular coordinate system. This forms the graph of $f$ based on the given polynomial or function expression $p(x)$.

Step 2: Graphing the function $g(x) = p(x) + q$ and describing the vertical shift

To graph $g(x) = p(x) + q$, take every point on the graph of $f(x)$ and shift it vertically by $q$ units. If $q > 0$, the shift is upwards; if $q < 0$, the shift is downwards. This results in a new graph that is identical in shape to $f(x)$ but is shifted vertically.

Step 3: Describing the relationship between the graphs of $f$ and $g$

The graph of $g(x) = p(x) + q$ is the graph of $f(x)$ shifted downwards by $10$ units. This vertical shift is due to the addition of the constant $q$ to $p(x)$, which does not alter the shape of the graph but only its position relative to the y-axis.

Final Answer:

The relationship between the graphs of $f$ and $g$ is that the graph of $g$ is a vertical shift of the graph of $f$ by $-10$ units. If $q$ is positive, the shift is upwards; if $q$ is negative, the shift is downwards. This demonstrates how adding a constant to a function affects its graph on a Cartesian coordinate system without altering the shape of the graph.

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